Dynamics From Forced Symmetry-Breaking

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Systems with symmetries arise frequently in applications. Such systems exhibit behavior that is not typical of nonsymmetric systems. Even though this behavior may be more complex than expected techniques from equivariant bifurcation theory permit one to analyze its origin and properties. However, in many applications the presence of symmetry is an idealization. It is important therefore to understand which aspects of the dynamics of the symmetric system are robust with respect to small symmetry-breaking imperfections. In many cases the loss of both discrete and continuous symmetries introduces global bifurcations into the dynamics and hence the possibility of chaotic time-dependence. This project proposes a study of such symmetry-breaking perturbations in several important classes of systems arising incontinuum mechanics (fluid flow, chemically reacting systems etc) with emphasis on identifying robust behavior both near threshold of a primary instability and further away from it. The former gives rise to low-dimensional systems with broken symmetry via center manifold reduction, while the latter require a study of partial differential equations. Special emphasis is placed on understanding the effects of breaking both reflection symmetry and translation invariance (both separately and simultaneously). Phenomena arising from the breaking of time-translation invariance due to parametric forcing and the loss of Galilean invariance will also be studied. The goal of the project is to gain a deep understanding of the limitations on the applicability of idealized models and to provide a description of the type of behavior that is typical of ''nearly-symmetric'' systems. The project aims at understanding the basic principles behind robust behavior. For example, understanding the effects of broken reflection symmetry is essential when studying the effects of a small inclination of an apparatus to the horizontal, and the effects of through-flow or rotation on transitions from one state to another. Problems of this type arise in microscopic situations (thin films on horizontal or inclined substrates) in geophysics (for example in oceanography) and everywhere in between. Similarly, understanding the trapping of wavelike disturbances by spatial inhomogemeities is of importance not only in catalysis where the inhomogeneities may be on atomic scale, and in the trapping of weather patterns by topography.